Abstract

We exhibit the information processing capabilities of the first few terms that arise in the amplitude expansion for resonant scattering in a medium with a delay nonlinearity (generalized volume hologram). We begin by showing how the physics of intensity dependent charge transport near a two-photon resonance gives both delayed quadratic and quartic nonlinearities. After reviewing the utility for matrix associative memories exhibited by the delayed quadratic nonlinearity (the ordinary Gabor hologram), we examine the role of the quartic nonlinearity, which is a fourth rank tensor. The symmetries of this tensor determine the information processing capabilities (via multilinear correlations) of the medium in an optical computing paradigm. We find multiple basins of stability, Jordan strings, and cycles as possible dynamic behaviors for the medium. We indicate how each corresponds to an information processing task: multiple basins to multiassociative memory, Jordan strings and cycles to chain and sequence memory and to group-invariant pattern recognition. We briefly indicate how branching processes may be implemented by the fourth rank mode-coupling tensor.

© 1990 Optical Society of America

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    [CrossRef]
  2. H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
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  3. W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).
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    [CrossRef]
  5. A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [CrossRef]
  6. D. M. Pepper, “Nonlinear Optical Phase Conjugation,” Opt. Eng. 21, 156–183 (1982).
  7. K. M. Johnson, M. Armstrong, L. Hesselink, J. W. Goodman, “The Multiple Multiple-Exposure Hologram,” Appl. Opt. 24, 4467–4472 (1985).
    [CrossRef] [PubMed]
  8. M. S. Cohen, “Design of a New Medium For Volume Holographic Information Processing,” Appl. Opt. 25, 2288–2294 (1986).
    [CrossRef] [PubMed]
  9. J. J. Hopfield, “Neural Networks and Physical Systems With Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554–2558 (1982).
    [CrossRef] [PubMed]
  10. J. A. Anderson, “Cognitive and Psychological Computation With Neural Models,” IEEE Trans. Syst. Man Cybern. SMC-13, 799–815 (1983).
    [CrossRef]
  11. T. Kohonen, Self Organization and Associative Memory (Springer-Verlag, New York, 1984).
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  13. B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative Holographic Memory With Feedback Using Phase-Conjugate Mirrors,” Opt. Lett. 11, 118–120 (1986).
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  14. Y. Owechko, “Optoelectronic Resonator Neural Networks,” Appl. Opt. 26, 5104–5111 (1987).
    [CrossRef] [PubMed]
  15. A. Yariv, S.-K. Kwong, “Associative Memories Based On Message-Bearing Optical Modes in Phase-Conjugate Resonators,” Opt. Lett. 11, 186–188 (1986).
    [CrossRef] [PubMed]
  16. K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Appl. Opt. 26, 5061–5076 (1987).
    [CrossRef] [PubMed]
  17. H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
    [CrossRef]
  18. C. L. Giles, T. Maxwell, “Learning, Invariance, and Generalization in High-Order Neural Networks,” Appl. Opt. 26, 4972–4978 (1987).
    [CrossRef] [PubMed]
  19. D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementation,” Neural Networks 1, 149–163 (1988).
    [CrossRef]
  20. Y. Owechko, G. J. Dunning, E. Marom, B. H. Soffer, “Holographic Associative Memory With Nonlinearities in the Correlation Domain,” Appl. Opt. 26, 1900–1910 (1987).
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    [CrossRef]
  23. A. C. Newell, “Bifurcation and Nonlinear Focusing,” in Pattern Formation and Pattern Recognition (Springer-Verlag, New York, 1980), pp. 244–265.
  24. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 618.
  25. P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
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  26. D. L. Hecht, “Multifrequency Acoustooptic Diffraction,” IEEE Trans Sonics Ultrason. SU-24, 7–18 (1977).
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    [CrossRef]
  28. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).
  29. A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).
  30. V. Guillemin, A. Pollock, Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974).
  31. S.N. Chow, J. Hale, Methods of Bifurcation Theory (Springer-Verlag, New York, 1982).
    [CrossRef]
  32. M. S. Cohen, “Multiple Correlations in a Holographic Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 122–131 (1988).
  33. F. John, Partial Differential Equations (Springer-Verlag, New York, 1982), p. 48.
  34. M. S. Cohen, “Multistability and Associative Memory in a Phase Conjugating Resonator,” J. Opt. Soc. Am. B, in press.
  35. D. Psaltis, J. Hong, S. Venkatesh, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189–195 (1986).
  36. H. Sompolinsky, I. Kanter, “Temporal Association in Asymmetric Neutral Networks,” Phys. Rev. Lett. 57, 2861–2864 (1986).
    [CrossRef] [PubMed]
  37. H. Kogelnik, C. V. Shank, “Stimulated Emission in a Periodic Structure,” Appl. Phys. Lett. 18, 152–154 (1970).
    [CrossRef]
  38. B. Noble, Applied Linear Algebra (Prentice-Hall, Englewood Cliffs, NJ, 1969).

1988 (2)

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementation,” Neural Networks 1, 149–163 (1988).
[CrossRef]

M. S. Cohen, “Multiple Correlations in a Holographic Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 122–131 (1988).

1987 (5)

1986 (6)

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

A. Yariv, S.-K. Kwong, “Associative Memories Based On Message-Bearing Optical Modes in Phase-Conjugate Resonators,” Opt. Lett. 11, 186–188 (1986).
[CrossRef] [PubMed]

B. H. Soffer, G. J. Dunning, Y. Owechko, E. Marom, “Associative Holographic Memory With Feedback Using Phase-Conjugate Mirrors,” Opt. Lett. 11, 118–120 (1986).
[CrossRef] [PubMed]

M. S. Cohen, “Design of a New Medium For Volume Holographic Information Processing,” Appl. Opt. 25, 2288–2294 (1986).
[CrossRef] [PubMed]

D. Psaltis, J. Hong, S. Venkatesh, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189–195 (1986).

H. Sompolinsky, I. Kanter, “Temporal Association in Asymmetric Neutral Networks,” Phys. Rev. Lett. 57, 2861–2864 (1986).
[CrossRef] [PubMed]

1985 (3)

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).

K. M. Johnson, M. Armstrong, L. Hesselink, J. W. Goodman, “The Multiple Multiple-Exposure Hologram,” Appl. Opt. 24, 4467–4472 (1985).
[CrossRef] [PubMed]

1984 (1)

1983 (1)

J. A. Anderson, “Cognitive and Psychological Computation With Neural Models,” IEEE Trans. Syst. Man Cybern. SMC-13, 799–815 (1983).
[CrossRef]

1982 (2)

D. M. Pepper, “Nonlinear Optical Phase Conjugation,” Opt. Eng. 21, 156–183 (1982).

J. J. Hopfield, “Neural Networks and Physical Systems With Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554–2558 (1982).
[CrossRef] [PubMed]

1979 (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

1978 (1)

A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

1977 (1)

D. L. Hecht, “Multifrequency Acoustooptic Diffraction,” IEEE Trans Sonics Ultrason. SU-24, 7–18 (1977).
[CrossRef]

1970 (1)

H. Kogelnik, C. V. Shank, “Stimulated Emission in a Periodic Structure,” Appl. Phys. Lett. 18, 152–154 (1970).
[CrossRef]

1969 (2)

A. C. Newell, J. A. Whitehead, “Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech. 38, 279–303 (1969).
[CrossRef]

D. Gabor, “Associative Holographic Memory,” IBM J. Res. Dev. 13, 156–159 (1969).
[CrossRef]

1968 (1)

H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972), p. 5.

Anderson, D. Z.

Anderson, J. A.

J. A. Anderson, “Cognitive and Psychological Computation With Neural Models,” IEEE Trans. Syst. Man Cybern. SMC-13, 799–815 (1983).
[CrossRef]

Armstrong, M.

Borshch, A.

Brodin, M.

Cathey, W. T.

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

Chen, H. H.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Chow, S.N.

S.N. Chow, J. Hale, Methods of Bifurcation Theory (Springer-Verlag, New York, 1982).
[CrossRef]

Cohen, M. S.

M. S. Cohen, “Multiple Correlations in a Holographic Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 122–131 (1988).

M. S. Cohen, “Design of a New Medium For Volume Holographic Information Processing,” Appl. Opt. 25, 2288–2294 (1986).
[CrossRef] [PubMed]

M. S. Cohen, “Multistability and Associative Memory in a Phase Conjugating Resonator,” J. Opt. Soc. Am. B, in press.

Dunning, G. J.

Gabor, D.

D. Gabor, “Associative Holographic Memory,” IBM J. Res. Dev. 13, 156–159 (1969).
[CrossRef]

Giles, C. L.

C. L. Giles, T. Maxwell, “Learning, Invariance, and Generalization in High-Order Neural Networks,” Appl. Opt. 26, 4972–4978 (1987).
[CrossRef] [PubMed]

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Goodman, J. W.

Guillemin, V.

V. Guillemin, A. Pollock, Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974).

Hale, J.

S.N. Chow, J. Hale, Methods of Bifurcation Theory (Springer-Verlag, New York, 1982).
[CrossRef]

Hecht, D. L.

D. L. Hecht, “Multifrequency Acoustooptic Diffraction,” IEEE Trans Sonics Ultrason. SU-24, 7–18 (1977).
[CrossRef]

Hesselink, L.

Hong, J.

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementation,” Neural Networks 1, 149–163 (1988).
[CrossRef]

D. Psaltis, J. Hong, S. Venkatesh, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189–195 (1986).

Hopfield, J. J.

J. J. Hopfield, “Neural Networks and Physical Systems With Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554–2558 (1982).
[CrossRef] [PubMed]

Huignard, H. P.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 618.

John, F.

F. John, Partial Differential Equations (Springer-Verlag, New York, 1982), p. 48.

Johnson, K. M.

Kanter, I.

H. Sompolinsky, I. Kanter, “Temporal Association in Asymmetric Neutral Networks,” Phys. Rev. Lett. 57, 2861–2864 (1986).
[CrossRef] [PubMed]

Khokhlov, R. V.

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972), p. 5.

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Stimulated Emission in a Periodic Structure,” Appl. Phys. Lett. 18, 152–154 (1970).
[CrossRef]

Kohonen, T.

T. Kohonen, Self Organization and Associative Memory (Springer-Verlag, New York, 1984).

Kukhtarev, N.

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

Kwong, S.-K.

Lee, H. Y.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Lee, Y. C.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Lininger, D. M.

Longuet-Higgins, H. C.

H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

Marom, E.

Marrakchi, A.

A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).

Maxwell, T.

C. L. Giles, T. Maxwell, “Learning, Invariance, and Generalization in High-Order Neural Networks,” Appl. Opt. 26, 4972–4978 (1987).
[CrossRef] [PubMed]

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Newell, A. C.

A. C. Newell, J. A. Whitehead, “Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech. 38, 279–303 (1969).
[CrossRef]

A. C. Newell, “Bifurcation and Nonlinear Focusing,” in Pattern Formation and Pattern Recognition (Springer-Verlag, New York, 1980), pp. 244–265.

Noble, B.

B. Noble, Applied Linear Algebra (Prentice-Hall, Englewood Cliffs, NJ, 1969).

Odulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

Owechko, Y.

Park, C. H.

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementation,” Neural Networks 1, 149–163 (1988).
[CrossRef]

Pepper, D. M.

D. M. Pepper, “Nonlinear Optical Phase Conjugation,” Opt. Eng. 21, 156–183 (1982).

Pollock, A.

V. Guillemin, A. Pollock, Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974).

Psaltis, D.

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementation,” Neural Networks 1, 149–163 (1988).
[CrossRef]

K. Wagner, D. Psaltis, “Multilayer Optical Learning Networks,” Appl. Opt. 26, 5061–5076 (1987).
[CrossRef] [PubMed]

D. Psaltis, J. Hong, S. Venkatesh, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189–195 (1986).

A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).

Rajbenbach, H.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, P.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Shank, C. V.

H. Kogelnik, C. V. Shank, “Stimulated Emission in a Periodic Structure,” Appl. Phys. Lett. 18, 152–154 (1970).
[CrossRef]

Soffer, B. H.

Solymar, C.

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Sompolinsky, H.

H. Sompolinsky, I. Kanter, “Temporal Association in Asymmetric Neutral Networks,” Phys. Rev. Lett. 57, 2861–2864 (1986).
[CrossRef] [PubMed]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

Starkov, V.

Sun, G. Z.

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Tanguay, A. R.

A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).

Venkatesh, S.

D. Psaltis, J. Hong, S. Venkatesh, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189–195 (1986).

Vinetski, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

Volkov, V.

Wagner, K.

Whitehead, J. A.

A. C. Newell, J. A. Whitehead, “Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech. 38, 279–303 (1969).
[CrossRef]

Yariv, A.

A. Yariv, S.-K. Kwong, “Associative Memories Based On Message-Bearing Optical Modes in Phase-Conjugate Resonators,” Opt. Lett. 11, 186–188 (1986).
[CrossRef] [PubMed]

A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yeh, P.

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

Yu, J.

A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).

AIP Conf. Proc. (1)

H. H. Chen, Y. C. Lee, G. Z. Sun, H. Y. Lee, T. Maxwell, C. L. Giles, “High-Order Correlation Model for Associative Memory,” AIP Conf. Proc. 151, 86–99 (1986).
[CrossRef]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

H. Kogelnik, C. V. Shank, “Stimulated Emission in a Periodic Structure,” Appl. Phys. Lett. 18, 152–154 (1970).
[CrossRef]

Ferroelectrics (1)

N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, V. L. Vinetski, “Holographic Storage in Electro-Optic Crystals,” Ferroelectrics 22, 949–961 (1979).
[CrossRef]

IBM J. Res. Dev. (1)

D. Gabor, “Associative Holographic Memory,” IBM J. Res. Dev. 13, 156–159 (1969).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, “Phase Conjugate Optics and Real-Time Holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

IEEE Trans Sonics Ultrason. (1)

D. L. Hecht, “Multifrequency Acoustooptic Diffraction,” IEEE Trans Sonics Ultrason. SU-24, 7–18 (1977).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

J. A. Anderson, “Cognitive and Psychological Computation With Neural Models,” IEEE Trans. Syst. Man Cybern. SMC-13, 799–815 (1983).
[CrossRef]

J. Appl. Phys. (1)

P. Refregier, C. Solymar, H. Rajbenbach, H. P. Huignard, “Two-Beam Coupling in Photorefractive Bi12SiO20 Crystals With Moving Grating: Theory and Experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Fluid Mech. (1)

A. C. Newell, J. A. Whitehead, “Finite Bandwidth, Finite Amplitude Convection,” J. Fluid Mech. 38, 279–303 (1969).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature London (1)

H. C. Longuet-Higgins, “Holographic Model of Temporal Recall,” Nature London 217, 104 (1968).
[CrossRef]

Neural Networks (1)

D. Psaltis, C. H. Park, J. Hong, “Higher Order Associative Memories and Their Optical Implementation,” Neural Networks 1, 149–163 (1988).
[CrossRef]

Opt. Eng. (2)

D. M. Pepper, “Nonlinear Optical Phase Conjugation,” Opt. Eng. 21, 156–183 (1982).

A. Marrakchi, A. R. Tanguay, J. Yu, D. Psaltis, “Physical Characteristics of the Photorefractive Incoherent to Coherent Optical Converter,” Opt. Eng. 24, 124–131 (1985).

Opt. Lett. (2)

Phys. Rev. Lett. (1)

H. Sompolinsky, I. Kanter, “Temporal Association in Asymmetric Neutral Networks,” Phys. Rev. Lett. 57, 2861–2864 (1986).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (1)

J. J. Hopfield, “Neural Networks and Physical Systems With Emergent Collective Computational Abilities,” Proc. Natl. Acad. Sci. U.S.A. 79, 2554–2558 (1982).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

D. Psaltis, J. Hong, S. Venkatesh, “Shift Invariance in Optical Associative Memories,” Proc. Soc. Photo-Opt. Instrum. Eng. 625, 189–195 (1986).

M. S. Cohen, “Multiple Correlations in a Holographic Resonator,” Proc. Soc. Photo-Opt. Instrum. Eng. 882, 122–131 (1988).

Other (11)

F. John, Partial Differential Equations (Springer-Verlag, New York, 1982), p. 48.

M. S. Cohen, “Multistability and Associative Memory in a Phase Conjugating Resonator,” J. Opt. Soc. Am. B, in press.

B. Noble, Applied Linear Algebra (Prentice-Hall, Englewood Cliffs, NJ, 1969).

V. Guillemin, A. Pollock, Differential Topology (Prentice-Hall, Englewood Cliffs, NJ, 1974).

S.N. Chow, J. Hale, Methods of Bifurcation Theory (Springer-Verlag, New York, 1982).
[CrossRef]

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

A. C. Newell, “Bifurcation and Nonlinear Focusing,” in Pattern Formation and Pattern Recognition (Springer-Verlag, New York, 1980), pp. 244–265.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), p. 618.

T. Kohonen, Self Organization and Associative Memory (Springer-Verlag, New York, 1984).

W. T. Cathey, Optical Information Processing and Holography (Wiley, New York, 1974).

S. A. Akhmanov, R. V. Khokhlov, Problems of Nonlinear Optics (Gordon & Breach, New York, 1972), p. 5.

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Figures (5)

Fig. 1
Fig. 1

Exposure phase. A volume of holographic medium is exposed to a superimposition of plane waves V s = A j s exp ( i k j s x ); each plane wave corresponds to a point p j s on the sth input pattern via the Fourier Transforming lens. Summing over M disjoint exposure phases s = 1,…,M, stores the multiplexed autocorrelation gratings κ = C s V s V s H and the fourth-order correlations T = D s V s V s H V s V s H. (In practice, the interfering waves are made to intersect at greater angles by a beam splitter and a system of mirrors32,34; these do not affect the analysis and hence are not shown here.) During the readout phase, a (perhaps distorted) portion V ^ J (0) of one of the stored s = J images is input at z = 0; V ^ J (0) serves as the initial condition for coupled wave Eqs. (21), giving the evolution of the scattered plane wave amplitudes with depth z in the medium.

Fig. 2
Fig. 2

A Bragg cone. (a) Wave vectors k l s and k m s interfere to form grating k l s - k m s. The Bragg conditions are fulfilled for any pair (kj,kp) on the Bragg cone k j - k p = k l s - k m s , k j = k p = k l s = k m s . (b) Each stored wave vector k j s lies on the n − 1 Bragg cones generated by interfacing k j s against all the other k l s; each accidental wave vector ka, however, lies generically on the intersection of at most two Bragg cones.

Fig. 3
Fig. 3

Energy surface (X1,…,XM) for EI ≠ 0 is a quartic surface with pockets in it along axes XI, I = 1,…,M, which represent stored vectors Vs after diagonalization ( is plotted here as the distance from the origin). Each pocket represents a basin of stability. An initially input vector V ¯ J (0) will relax into the basin of attraction VJ nearest it, i.e., for which 〈Vs,VJ(0)〉 was maximal. The energy surface for EI = 0, however, is a hypersphere; any vector from the origin landing on the hypersphere represents a neutrally stable equilibrium.

Fig. 4
Fig. 4

Numerical solution to Eq. (23) with ρ = 0, all CI = 2, g = 2, all EI = 1. The horizontal axis is index I = 1,…,10. The vertical axis is amplitude XI(T). Plots are drawn for fixed τ. With a random input vector, the solution converges to 2 · ( 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ), a multiple of the stored vector V3 most resembling the input.

Fig. 5
Fig. 5

Numerical solution to Eq. (25) with ρ = 0, all C ^ I = C I = 2, g = 2, all EI = 1, p = 0.40. The input of V10 produces sequential reconstruction of the string V9,V8,V7,V6,… V3. When the calculation is continued, the string terminates in V1 as the asymptotically stable steady state. The system dwells for awhile in each basin Vs+1 then jumps rapidly to Vs. Other computations show that the process cycles if V 10 V 1 H is also stored.

Tables (1)

Tables Icon

Table I Mapping of Information Processing Tasks into Coupled Wave Dynamics

Equations (59)

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2 E - μ 0 ɛ 0 E t t = μ 0 P t t .
P ( E ) = χ 1 E + χ 2 E 2 + χ 3 E 3 + 0 ( E 5 ) ,
n T = - β n + D Δ n + F ( I ) ,
F ( I ) α 1 E 2 + α 2 E 4 ,
n ( T , x ) = r = 1 n n r ( T ) exp ( i q r · x ) + c . c . ,
E ( T , x ) = j = 1 n A j ( T ) exp ( i k j · x ) + c . c . ,
n ( T , x ) = α 1 ( j l { 0 T exp [ - ( β + D k j - k l ) ( T - T ) ] × A j A ¯ l ( T ) d T } exp [ i ( k j - k l ) · x ] + c . c . ) + α 2 ( j k l m { 0 T exp [ - ( β + D k j - k k + k l - k m 2 ) × ( T - T ) ] A j A ¯ k A l A ¯ m ( T ) d T } exp [ i ( k j - k k + k l - k m ) · x ] + c . c . ) .
χ 1 = χ 0 , 1 + α χ 2 , 1 { E 2 } + α χ 4 , 1 { E 4 } ,
χ 2 , 1 { E 2 } = Λ 2 , 1 j l { 0 T exp [ - β ( T - T ) ] A j A ¯ l ( T ) d T } × exp [ i ( k j - k l ) · x ] + c . c .
χ 4 , 1 { E 4 } = Λ 4 , 1 j k l m { 0 T exp [ - β ( T - T ) ] A j A ¯ k A l A ¯ m ( T ) d T } × exp [ i ( k j - k k + k l - k m ) · x ] + c . c .
χ 3 = α χ 0 , 3 + α χ 2 , 3 { E 2 } + α χ 4 , 3 { E 4 } ,
χ 2 , 3 { E 2 } = Λ 2 , 3 j l { 0 T exp [ - β ( T - T ) ] A j A ¯ l ( T ) d T } × exp [ i ( k j - k l ) · x ] + c . c .
χ 4 , 3 { E 4 } = Λ 4 , 3 j k l m { 0 T exp [ - β ( T - T ) ] A j A ¯ k A l A ¯ m ( T ) d T } × exp [ i ( k j - k k + k l - k m ) · x ] + c . c .
P { E } = χ 0 , 1 E + α [ χ 2 , 1 { E 2 } E + χ 4 , 1 { E 4 } E + χ 0 , 3 E 3 + χ 2 , 3 { E 2 } E 3 + χ 4 , 3 { E 4 } E 3 + O ( E 5 ) ] ,
χ 2 , 1 { E 2 } = Λ 2 , 1 [ s = 1 M n j l n T s - 1 T s exp [ - β ( T - T ) ] A j s A ¯ l s d T ] × exp [ i ( k j s - k l s ) · x ] + c . c . = s = 1 M c s ( T ) n j l n A j s A ¯ l s exp [ i ( k j s - k l s ) · x ] + c . c . ,
c s ( T ) = Λ 2 , 1 β exp ( - β T ) [ exp ( β T s ) - exp ( β T s - 1 ) ]
χ 2 , 3 { E 2 } = s m n l m n d s ( T ) A l s A ¯ m s exp [ i ( k l s - k m s ) · x ] + c . c . ,
χ 4 , 3 { E 4 } = s M n k l n n m w n e s ( T ) A k s A ¯ l s A m s A ¯ w s × exp [ i ( k k s - k l s + k m s - k w s ) · x ] + c . c . ,
E ( T , t , x , X ) = j = 1 N A j ( T , X ) exp ( i k j · x ) exp ( i ω t ) ,
( k l s - k m s ) + k p = k j ,             l m
( k l s - k m s ) + ( k p - k q + k r ) = k j             ( for l m ) .
( k k s - k l s + k m s - k w s ) + ( k p - k q + k r ) = k j             ( for k l , m w ) .
d d ξ j A j ( T - v k ^ j · X ) A j = - ρ A j + ( s = 1 M C s m = 1 n A j s A ¯ m s ) A m - g ( p = 1 n A p 2 ) A j + ( s = 1 M D s l = 1 n m = 1 n A l s A ¯ m s A ¯ l A m ) A j + ( s = 1 M E s j = 1 n k = 1 n l = 1 n m = 1 n A j s A ¯ k s A l s A ¯ m s ) A k A ¯ l A m + O ( A 5 ) ,
ρ = v 2 μ 0 σ ω ,             g = v μ 0 ω 2 exp ( i γ ) χ 0 , 3 , C s = v 2 μ 0 ω 2 exp ( i θ ) ( c s ) ,             D s = v μ 0 ω 2 exp ( i ψ ) ( d s ) , E s = 2 v μ 0 ω 2 exp ( i φ ) ( e s ) .
ξ j = T - 1 v k ^ j · X .
k ^ j · X A j d d z A j ,
d A j d τ = - ρ A j + m = 1 n κ j m A m - g A j ( p = 1 n A p 2 ) + A j l = 1 n m = 1 n R l m A ¯ l A m + k = 1 n l = 1 n m = 1 n T j k l m A k A ¯ l A m ,
κ j m = s = 1 M C s A j s A ¯ m s ,
R l m = s = 1 M D s A l s A ¯ m s ,
T j k l m = s = 1 M E s A j s A ¯ k s A l s A ¯ m s .
V ( τ ) = [ A 1 ( τ ) A N ( τ ) ] ,             V S = [ A 1 s A n s ]
V s , V = 1 Ω Ω [ j = 1 N A ¯ j s exp ( - i k j s · x ) j = 1 N A j exp ( i k j · x ) ] d Ω ,
d d τ V = - ρ V + κ V + ( V H R V ) V + T ( V , V H , V ) - g V 2 V ,
κ = C s V s V s H ,
R = D s V s V s H ,
T = E s V s V s H V s V s H ,
d d τ V = - ρ V + C s V s , V V s + ( D s V s , V 2 ) V + E s V s , V 2 V s , V V s - g V 2 V .
d d τ X I = ( C I - ρ ) X I + ( s = 1 M D s X s 2 ) X I + E I X I 2 X I - g ( J = 1 n X J 2 ) X I
d d τ X I = H X ¯ I ,
H = J = 1 n ( C J - ρ ) X J 2 - g 2 ( J = 1 n X J 2 ) 2 + 1 2 J = 1 n E J X J 4 2 [ G + i H ] .
d d τ q I = G q I ,             d d τ p I = G p I ,
X I 2 = C I - σ g - E I ,             X J = 0 for J I .
d d τ q I = H p I ,             d d τ p I = - H q I ,
X I 2 = Re ( C I - σ ) Re ( g - E I ) , X J = 0 for J I ,
κ = C s V s V s H .
κ = C s ( U s + W s ) ( U s H + W s H )
κ ^ = s = 1 M - 1 C s V s V s + 1 H .
κ ^ = s = 1 M - 1 C s V s V s + 1 H 1 2 π s C s × 0 2 π [ V s + exp ( i θ ) V s + 1 ] [ V s + exp ( i θ ) V s + 1 ] H exp ( i θ ) d θ .
d d τ X I = [ - ρ + C ^ I ( 1 - p ) ] X I + C I p X I + 1 - g X I X l 2 + E I X I 2 X I
E ( x , X , t , T ) = p = 1 N A p ( X , T ) exp ( i k p · x ) exp ( i ω t ) ,
p = 1 N A p ( T - v k ^ p · X ) exp ( i k p · x ) = - v [ μ 0 σ ω 2 ] p = 1 N A p exp ( i k p · x ) + v μ 0 ω 2 2 [ exp ( i θ ) { s = 1 M c s n l m n A l s A ¯ m s exp [ i ( k l s - k m s ) · x ] + c . c . } p = 1 N A p exp ( i k p · x ) + ( χ 0 , 3 + exp ( i ψ ) { s = 1 M d s n l m n A l s A ¯ m s exp [ i ( k l s - k m s ) · x ] + c . c . } + exp ( i φ ) { s = 1 M e s n k l n n m w n A k A ¯ l s A m s A ¯ w s exp [ i ( k k s - k l s + k m s - k w s ) · x ] + c . c . } + ) p = 1 N q = 1 N r = 1 N A p A ¯ q A r exp [ i ( k p - k q + k r ) · x ] ] + O ( A 5 ) ,
d d τ ( T - v k ^ j · X )
( k l s - k m s ) + k p = k j ,
d d τ [ V W ] = [ κ - ρ I 0 0 B - ρ I ] [ V W ] ,
κ j l = s = 1 m C s A j s A ¯ l s ,             except κ j j = 0 ,
C s = v 2 μ 0 ω 2 exp ( i θ ) c s .
α j ( γ ) + ρ max ( κ l m + κ l m ) 2 max ( κ l m ) ,
s = 1 M A j s A l s < 1 2 .
A j s < 1 2 M             for all j = 1 , n , s = 1 , , M ,

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